New floor/subfloor in kitchen. Need ideas

I am remodeling my kitchen, and finally stripped some of the laminate(?) flooring off to reveal the hardwood underneath, only to find that it is in pretty bad shape in some areas. My house has hardwood floors, and NO SUBFLOOR.

What would you do with this room? My thoughts were to remove the hardwood flooring, install 3/4"(thick enough?) plywood, then put cement backer down, and then ceramic tile. What do you guys think about this?

ALSO, I have a breakfast nook adjacent to this room that I also need ideas for. It currently has two layers on vinyl tiles on it, that were not coming up very easily. I have a couple of ideas here

1) remove the tiles + hardwood/subfloor, and do the same as in the kitchen(plywood, backerboard, tile). This seems kind of like overkill since the hardwood floors are in good shape here

2) remove the tiles + refinish the hardwood. The hardwood was nice under those floors w/ cool inlays and what not. I just don't know how well it will clean up. Plus, if asbestos glue was used to hold the tiles down, I would rather not go sanding that up.

3) remove the tiles + keep the hardwood + install cement backer over this and tile. Though this might be uneven w/ the kitchen because the hardwood is probably an inch thick.